Ordinal association
Kendall's Tau Calculator — tau-b & tau-c
Kendall's tau measures ordinal association between two variables from concordant and discordant pairs; tau-b corrects for ties (best for square tables) and tau-c adjusts for table shape (best for rectangular tables). Both range from −1 to +1.
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Open the workspace →What is Kendall's tau?
Kendall's tau is a rank-correlation measure for two ordinal variables. It looks at every pair of observations and asks whether they are concordant (one case ranks higher on both variables) or discordant (it ranks higher on one but lower on the other). An excess of concordant pairs produces a positive tau; an excess of discordant pairs produces a negative one.
The wrinkle is ties — pairs that share the same row category, the same column category, or both. Plain measures ignore them. Tau-b folds tied pairs back into the denominator so the measure reaches ±1 on square tables, while tau-c rescales using the table shape so it behaves well on rectangular tables.
Formula
Definition
tau-b = (C − D) / √((C + D + Tr)(C + D + Tc))
- C, D
- = the number of concordant and discordant pairs
- Tr, Tc
- = pairs tied on the row variable and on the column variable, respectively
- tau-b
- = best suited to square tables (equal numbers of rows and columns)
- tau-c = 2m(C − D) / (n²(m − 1))
- = Stuart's tau-c, where m = min(rows, columns) and n is the grand total; best suited to rectangular (non-square) tables
Worked example
Worked example
Consider this ordinal 3×3 table (both variables run low → high):
[[20, 10, 5], [10, 20, 10], [5, 10, 20]]
Counting across every pair of cases, concordant pairs (higher on both variables) clearly outnumber discordant pairs (higher on one, lower on the other), so tau is positive. After folding the row and column ties into the denominator for tau-b — and rescaling by the table shape for tau-c — the measures come out to:
tau-b = 0.391
tau-c = 0.390
Both indicate a moderate positive ordinal association: as one variable increases, the other tends to increase too. The two values are nearly identical here because the table is square.
When to use it
Use it when
- Both variables are ordinal (ordered categories).
- The table is square — use tau-b, which corrects for ties.
- The table is rectangular (unequal rows and columns) — use tau-c, which adjusts for table shape.
- You want a measure that properly handles ties, unlike gamma.
Not the right tool when
- The data are nominal (unordered) — use Cramér's V instead.
- You have a very large n and only need a quick effect size — a simpler summary may suffice.
How to interpret it
Rule of thumb
Tau ranges from −1 to +1: the sign gives the direction of association and the magnitude gives its strength. Tau is usually smaller in magnitude than gamma because it accounts for ties rather than discarding them.
Frequently asked questions
- tau-b vs tau-c — which should I use?
- Use tau-b for square tables (the same number of rows and columns), since it corrects for ties and can reach ±1. Use tau-c for rectangular tables, since it adjusts for the table's shape so the measure is not artificially capped below 1. On a square table the two are nearly identical.
- Kendall's tau vs gamma?
- Goodman and Kruskal's gamma ignores tied pairs entirely, so it tends to be larger in magnitude. Kendall's tau keeps ties in the denominator, giving a more conservative measure. If your data have many ties, tau is usually the more honest summary of association.
- What does a tau of 0.39 mean?
- A tau of about 0.39 indicates a moderate positive ordinal association: as one variable increases, the other tends to increase as well, but not perfectly. The positive sign gives the direction and the magnitude (well short of 1) gives the moderate strength.
References & further reading
- Kendall, M. G. (1938). A New Measure of Rank Correlation. Biometrika, 30(1/2), 81–93.
- Stuart, A. (1953). The Estimation and Comparison of Strengths of Association in Contingency Tables. Biometrika, 40(1/2), 105–110.
- Kendall rank correlation coefficient — Wikipedia
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